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Math Chapter 7 : Topic 6
Distance and Midpoint Applications
1 / 25
Two towns are located at $(2,3)$ and $(6,7)$ on a map. The distance between them represents:
Reasoning: Distance formula gives shortest straight-line path.
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A delivery point is exactly midway between $(2,4)$ and $(6,8)$. Its coordinates are:
Reasoning: $M$ $\left(\frac{x_{1} + x_{2}}{2},\frac{y_{1} + y_{2}}{2}\right) ;$Midpoint gives central location.
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Distance between two cities at $(0,0)$ and $(3,4)$ is:
Reasoning: $d=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}=\sqrt{3^{2}+4^{2}}=5$
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A park lies midway between two houses at $(1,5)$ and $(5,5)$. Its position is:
Reasoning: Average $x$, same $y$.
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Two points are $10$ units apart horizontally. Their vertical difference is zero. The distance is:
Reasoning: Only horizontal distance matters.
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A bridge is to be built exactly halfway between $(- 2,2)$ and $(6,6)$. he location is:
Reasoning: Midpoint formula$⟹ M$ $\left(\frac{x_{1} + x_{2}}{2},\frac{y_{1} + y_{2}}{2}\right)$
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Distance between $(2,3)$ and $(2,9)$ represents:
Reasoning: Same $x$-coordinate
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A rescue team is at $(0,0)$. A victim is at $(6,8)$. Distance to travel is:
Reasoning: $d=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}=\sqrt{36+64}=10$
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Midpoint is useful in real life to:
Reasoning: Used to locate center between two points.
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Distance formula gives:
Reasoning: Straight-line (shortest distance).
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A road connects $(1,2)$ and $(5,6)$. Its length is:
Reasoning: $d=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}=\sqrt{(4)^{2}+(4)^{2}}=\sqrt{32}$
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If midpoint of two locations is $(3,3)$, one location is $(1,1)$, the other is:
Reasoning: Solve midpoint equations.Midpoint formula$⟹ M$ $\left(\frac{x_{1} + x_{2}}{2},\frac{y_{1} + y_{2}}{2}\right)$
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Distance between $(- 3, - 4)$ and $(0,0)$ is:
Reasoning: $d=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}$
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A midpoint divides a road into:
Reasoning: Midpoint splits equally.
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Two towers are at $(2,2)$ and $(8,2)$. The midpoint is:
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Distance between two identical locations is:
Reasoning: Same point$⟹$no distance
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A drone flies from $(0,0)$ to $(5,12)$. Distance traveled is:
Reasoning: $d=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}=\sqrt{25+144}=13$
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Midpoint of $(- 4,6)$ and $(4,- 6)$ is:
Reasoning: Symmetry cancels values.Midpoint formula$⟹ M$ $\left(\frac{x_{1} + x_{2}}{2},\frac{y_{1} + y_{2}}{2}\right)$
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A map scale uses coordinate points. Distance formula helps to:
Reasoning: Converts coordinate difference into measurable distance.
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If two points are far apart diagonally, distance formula is preferred over:
Reasoning: Diagonal distance requires full formula.
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Midpoint of $(0,6)$ and $(6,0)$ represents:
Reasoning: It’s the balanced central location.
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Distance between $(1,1)$ and $(4,5)$ is used to find:
Reasoning: Distance formula measures path.
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If midpoint is used in construction, it ensures:
Reasoning: Midpoint ensures symmetry.
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A midpoint lies:
Reasoning: Always between two points.
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Distance formula is derived from:
Reasoning: Based on right triangle concept.
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